For problem (1), there exists a locally minimizing singular arc S defined in (p, x)-space by:
Let S_ denote the integral curve of [[SIGMA].sub.a] through the upper saturation point ([p.sup.*.sub.u], [x.sup.*.sub.u]) of the singular arc for t < 0 until the value x = [square root of b/d] is reached and let [S.sub.+] denote the integral curve of [[SIGMA].sub.a] through the lower saturation point ([p.sup.*.sub.l], [x.sup.*.sub.l])f or t [greater than or equal to] 0.
It will only switch if the singular arc S is reached before the overall amount of drug is exhausted i.e.
The optimal controls are then referred to as singular arcs. The singular arcs [u.sub.1s] and [u.sub.2s] are then determined by equating the time derivatives to zero and solving.
The optimal controls for use of clean planting materials ([u.sub.1]) and debudding ([u.sub.2]) tend to suggest existence of singular arcs. In this case, the controls are implemented whenever the resources are available which better suits application than the bang-bang case where they are just switched between the lower and the upper limit.
Sebag, The Drinfeld-Grinberg-Kazhdan theorem is false for singular arcs
, To appear in Journal of IMJ (DOI: http://dx.doi.org/10.1017/S1474748015000341).